## Electronic Journal of Statistics

### Change-point detection in the marginal distribution of a linear process

#### Abstract

The subject of this paper is the detection of a change in the marginal distribution of a stationary linear process. By considering the marginal distribution, the change-point model can simultaneously incorporate any change in the coefficients and/or the innovations of the linear process. Furthermore, the change point can be random and data dependent. The key is an analysis of the asymptotic behaviour of the sequential empirical process, both with and without a change point. Our results hold under very mild conditions on the existence of any moment of the innovations and a corresponding condition of summability of the coefficients.

#### Article information

Source
Electron. J. Statist., Volume 10, Number 2 (2016), 3945-3985.

Dates
First available in Project Euclid: 13 December 2016

https://projecteuclid.org/euclid.ejs/1481598074

Digital Object Identifier
doi:10.1214/16-EJS1215

Mathematical Reviews number (MathSciNet)
MR3581958

Zentralblatt MATH identifier
1353.62095

Subjects
Secondary: 62G10, 62G30, 60F17

#### Citation

El Ktaibi, Farid; Ivanoff, B. Gail. Change-point detection in the marginal distribution of a linear process. Electron. J. Statist. 10 (2016), no. 2, 3945--3985. doi:10.1214/16-EJS1215. https://projecteuclid.org/euclid.ejs/1481598074

#### References

• [1] Abdelrazeq, I., Ivanoff, B. G. and Kulik, R. (2014). Model verification for Lévy-driven Ornstein-Uhlenbeck processes., Electronic Journal of Statistics. 8 1029–1062.
• [2] Berkes, I., Gombay, E. and Horváth, L. (2009). Testing for changes in the covariance structure of linear processes., Journal of Statistical Planning and Inference. 139 2044–2063.
• [3] Bickel, P. J. and Wichura, M. J. (1971). Convergence criteria for multiparameter stochastic processes and some applications., The Annals of Mathematical Statistics. 5 1656–1670.
• [4] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New, York.
• [5] Doukhan, P. and Surgailis, D. (1998). Functional central limit theorem for the empirical process of short memory linear processes., C.R. Acad. Sci. Paris. 326 87–92.
• [6] El Ktaibi, F. (2015). Asymptotics for the sequential empirical process and testing for distributional change for stationary linear models. Ph.D. thesis, University of Ottawa., http://hdl.handle.net/10393/31916
• [7] El Ktaibi, F., Ivanoff, B. G. and Weber, N. C. (2014). Bootstrapping the empirical distribution of a linear process., Statistics and Probability Letters. 93 134–142.
• [8] Giraitis, L., Leipus, R. and Surgailis, D. (1996). The change-point problem for dependent observations., Journal of Statistical Planning and Inference. 53 297–310.
• [9] Gordin, M. I. (1969). On the central limit theorem for stationary processes., Sov. Math. Dokl. 10 1174–1176.
• [10] Gorodetskii, V. V. (1977). On the strong mixing property for linear sequences., Theory Probab. Appl. 22 411–413.
• [11] Inoue, A. (2001). Testing for the distributional change in time series., Econometric Theory. 17 156–187.
• [12] Ivanoff, B. G. (1980). The function space $D([0,\infty)^q,E)$., The Canadian Journal of Statistics. 8 179–191.
• [13] Ivanoff, B. G. and Weber, N. C. (2010). Asymptotic results for spatial causal ARMA models., Electronic Journal of Statistics. 4 15–35.
• [14] McLeish, D. L. (1974). Dependent central limit theorems and invariance principles., Ann. Probab. 2 620–628.
• [15] Peligrad, M. (1998). On the blockwise bootstrap for empirical processes for stationary sequences., The Annals of Probability 26(2) 877–901.
• [16] Shao, X. and Zhang, X. (2010).Testing for change points in time series., Journal of the American Statistical Association, Theory and Methods 105 1228–1240.